CAT 2025 Lesson : Percentages - To & By and Multiplication Factor

In a fraction, the numerator is increased by $120\%$ while the denominator is increased to $160\%$, the resulting fraction is $\dfrac{1}{2}$. Which of the following is the fraction?

(1) $\dfrac{8}{22}$            (2) $\dfrac{13}{22}$            (3) $\dfrac{5}{6}$            (4) $\dfrac{2}{3}$           

Solution

Note that the numerator increases by $120\%$, while denominator increases to $160\%$.

Let the number be $\dfrac{x}{y}$.

$\therefore \dfrac{2.2 x}{1.6 y} = \dfrac{1}{2}$

⇒ $\dfrac{x}{y} = \dfrac{4}{11} = \dfrac{8}{22}$

Answer: (1) $\dfrac{8}{22}$

Population of a village increased by $60\%$ in $2001$, reduced to $75\%$ in $2002$, increased to $150\%$ in $2003$. If the population at the end of $2003$ was $54$ crores, then what was the population of the village in $2000$?

Solution

Note that by and to have been used for different years.

Let the population at the beginning of $2000$ be $x$.

$x \times 1.6 \times 0.75 \times 1.5 = 54$

These decimals can easily be expressed as fractions, which would ease the calculation.

⇒ $x \times \dfrac{8}{5} \times \dfrac{3}{4} \times \dfrac{3}{2} = 54$

⇒ $\dfrac{x \times 9}{5} = 54$

⇒ $x = 30$

Answer: $30$ crores

Cicero had only two forms of expenses - rent and food. In $2020$, food expense accounted for $60\%$ of his total expense. In $2021$, if his rent expense increased by $25\%$ and food expense reduced by $20\%$, then what was the percentage change in his total annual expense

Solution

Note that absolute values (in rupees) is not provided and we are required to find the percentage change. So, we can assume values. Using 100 or multiples of 100 makes calculations easier.

Secondly, we have 2 time periods (2020 and 2021) and the same type of expenses (rent and food).
Therefore, we can use a table to fill details. This improves the speed in solving, as well.

Let the expense in 2020 be Rs. 100.
∴ Food expense in 2020 is 60% of Rs. 100 = Rs. 60
Rent expense in 2020 = 100 – 60 = Rs. 40

Expense	2020	2021
Rent	40
Food	60
Total	100

Rent expense in 2021 = $\left( 1 + \dfrac{25}{100} \right) \times 40 = \dfrac{5}{4} \times 40$ = Rs. 50

Food expense in 2021 = $\left( 1 - \dfrac{20}{100} \right) \times 60 = \dfrac{4}{5} \times 60$ = Rs. 48

Expense	2020	2021
Rent	40	50
Food	60	48
Total	100	98

As expense has reduced from Rs. $100$ to Rs. $98$, there has been a decrease of $2\%$ in the total annual expense.

Answer: 2%

Kamla's height was $4$ feet $2$ inches last month. If she is $4.5$ feet now, calculate the percentage increase in her height?

Solution

$1$ foot = $12$ inches

$4$ feet $2$ inches = $48 + 2 = 50$ inches

$4.5$ feet = $4.5 \times 12 = 54$ inches

Change in height = $\dfrac{54 - 50}{50} \times$ 100% = 8%

Answer: $8\%$

Time taken to travel from Mecca to Medina used to be $8$ hours. Due to improved roads, the time taken has reduced by $37.5\%$. Calculate the current travel time.

Solution

$r = 37.5\% = \dfrac{3}{8}$

Current time taken $= 8 \times \left( 1 - \dfrac{3}{8} \right) = 8 \times \dfrac{5}{8} = 5$ hours

Answer: $5$ hours